Come on then @EnolaGaia
- put us out of our misery.
I got as far (on my own) as concluding the illusion of multiple shapes was a result of the thickness and contours of the grid's cells' "walls".
Then I found an explanation that confirmed the direction I was headed.
Look at the initial static image and notice how the grid is casting shadows underneath its structure - insinuating the structure can't be sitting flat on the table-top. Also note that in this initial image some of the circular cells show slight 'corners' (not-quite-circular points) around their peripheries.
The grid's cells are differently shaped from one side to the other, and whether you see pure circular versus more linear cell shapes depends on both (a) which side you're seeing and (b) the exact angle at which you're seeing the grid.
The illusion is a combination of different shapes on different sides and a bit of forced perspective to accentuate the different impressions each side can give.